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Language	Haskell98

Data.Eigen.LA

Contents

Basic linear solving
Multiple linear regression

Description

The problem: You have a system of equations, that you have written as a single matrix equation

Ax = b

Where A and b are matrices (b could be a vector, as a special case). You want to find a solution x.

The solution: You can choose between various decompositions, depending on what your matrix A looks like, and depending on whether you favor speed or accuracy. However, let's start with an example that works in all cases, and is a good compromise:

import Data.Eigen.Matrix
import Data.Eigen.LA

main = do
    let
        a = fromList [[1,2,3], [4,5,6], [7,8,10]]
        b = fromList [[3],[3],[4]]
        x = solve ColPivHouseholderQR a b
    putStrLn "Here is the matrix A:" >> print a
    putStrLn "Here is the vector b:" >> print b
    putStrLn "The solution is:" >> print x

produces the following output

Here is the matrix A:
Matrix 3x3
1.0 2.0 3.0
4.0 5.0 6.0
7.0 8.0 10.0

Here is the vector b:
Matrix 3x1
3.0
3.0
4.0

The solution is:
Matrix 3x1
-2.0000000000000004
1.0000000000000018
0.9999999999999989

Checking if a solution really exists: Only you know what error margin you want to allow for a solution to be considered valid.

You can compute relative error using norm (ax - b) / norm b formula or use relativeError function which provides the same calculation implemented slightly more efficient.

Synopsis

Basic linear solving

data Decomposition Source

Decomposition           Requirements on the matrix          Speed   Accuracy

PartialPivLU            Invertible                          ++      +
FullPivLU               None                                -       +++
HouseholderQR           None                                ++      +
ColPivHouseholderQR     None                                +       ++
FullPivHouseholderQR    None                                -       +++
LLT                     Positive definite                   +++     +
LDLT                    Positive or negative semidefinite   +++     ++
JacobiSVD               None                                -       +++

The best way to do least squares solving for square matrices is with a SVD decomposition (JacobiSVD)

Constructors

PartialPivLU	LU decomposition of a matrix with partial pivoting.
FullPivLU	LU decomposition of a matrix with complete pivoting.
HouseholderQR	Householder QR decomposition of a matrix.
ColPivHouseholderQR	Householder rank-revealing QR decomposition of a matrix with column-pivoting.
FullPivHouseholderQR	Householder rank-revealing QR decomposition of a matrix with full pivoting.
LLT	Standard Cholesky decomposition (LL^T) of a matrix.
LDLT	Robust Cholesky decomposition of a matrix with pivoting.
JacobiSVD	Two-sided Jacobi SVD decomposition of a rectangular matrix.

Instances

Enum Decomposition
Show Decomposition

solve :: Decomposition -> Matrix -> Matrix -> Matrix Source

x = solve d a b: finds a solution x of ax = b equation using decomposition d

relativeError :: Matrix -> Matrix -> Matrix -> Double Source

e = relativeError x a b: computes norm (ax - b) / norm b where norm is L2 norm

Multiple linear regression

linearRegression :: [[Double]] -> ([Double], Double) Source

(coeffs, error) = linearRegression points: computes multiple linear regression y = a1 x1 + a2 x2 + ... + an xn + b using ColPivHouseholderQR decomposition

point format is [y, x1..xn]
coeffs format is [b, a1..an]
error is calculated using relativeError

import Data.Eigen.LA
main = print $ linearRegression [
    [-4.32, 3.02, 6.89],
    [-3.79, 2.01, 5.39],
    [-4.01, 2.41, 6.01],
    [-3.86, 2.09, 5.55],
    [-4.10, 2.58, 6.32]]

produces the following output

([-2.3466569233817127,-0.2534897541434826,-0.1749653335680988],1.8905965120153139e-3)